The Annals of Statistics

Asymptotic Expansions in Anscombe's Theorem for Repeated Significance Tests and Estimation after Sequential Testing

Hajime Takahashi

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Abstract

Let $x_1, x_2, \cdots$ be independent and normally distributed with unknown mean $\theta$ and variance 1. Let $\tau = \inf \{n \geq 1: |s_n| \geq \sqrt{2a(n + c)}\}$. Then a repeated significance test for a normal mean rejects the hypothesis $\theta = 0$ if and only if $\tau \leq N_0$ for some positive integer $N_0$. The problem we consider is estimation of $\theta$ based on the data $x_1,\cdots, x_T, T = \min\{\tau, N_0\}$. We shall solve this problem by obtaining the asymptotic expansion of the distribution of $(s_\tau - \tau\theta)/\sqrt{\tau}$ as $a \rightarrow \infty$, and then constructing the confidence intervals for $\theta$.

Article information

Source
Ann. Statist., Volume 15, Number 1 (1987), 278-295.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176350266

Digital Object Identifier
doi:10.1214/aos/1176350266

Mathematical Reviews number (MathSciNet)
MR885737

Zentralblatt MATH identifier
0615.62107

JSTOR
links.jstor.org

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 62L12: Sequential estimation 60K05: Renewal theory

Keywords
Sequential estimation sequential test confidence interval Anscombe's theorem

Citation

Takahashi, Hajime. Asymptotic Expansions in Anscombe's Theorem for Repeated Significance Tests and Estimation after Sequential Testing. Ann. Statist. 15 (1987), no. 1, 278--295. doi:10.1214/aos/1176350266. https://projecteuclid.org/euclid.aos/1176350266


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